Size: px
Start display at page:

Download ""

Transcription

1 A Structure for General and Specc Market Rsk Eckhard Platen 1 and Gerhard Stahl Summary. The paper presents a consstent approach to the modelng of general and specc market rsk as dened n regulatory documents. It compares the statstcally based beta-factor model wth a class of benchmark models that use a broadly based ndex as major buldng block for modelng. The nvestgaton of log-return of stock prces that are expressed n unts of the market ndex reveals that these are lkely to be Student t dstrbuted. A correspondng dscrete tme benchmark model s then used to calculate Value-at-Rsk for equty portfolos. Key words: Rsk measurement, general market rsk, specc market rsk, Value at Rsk, nancal modelng, benchmark model, growth optmal portfolo. 1 Introducton Tradng portfolos of nancal nsttutons are characterzed by non-lnear nstruments, ted to complex tradng strateges. The nomnal volume of such postons s n general not proportonal to the rsk that s taken. Fnancal nsttutons can run nternal models for calculatng regulatory captal, see Basle (1996a, 1996b). These models provde forecast dstrbutons of portfolo losses due to uctuatons of market prces. In ths context t s mportant to see how regulatory terms are translated nto quanttatve rsk modelng. Market rsk, whch s due to uctuatons of market prces, plays an essental role n determnng regulatory captal. It s understood as the core rsk that an nsttuton s exposed to through ts tradng portfolo. Market rsk s splt nto general and specc market rsk. More precsely, for an equty portfolo general market rsk denotes the rsk exposure of the portfolo aganst the equty market as a whole. On the other hand, specc market rsk relates to the rsk of holdng an ndvdual securty wthn an equty portfolo, whch s not covered by general market rsk. Specc market rsk can be decomposed 1 Unversty oftechnology Sydney, School of Fnance & Economcs and Department of Mathematcal Scences, PO Box 13, Broadway, NSW, 007, Australa German Fnancal Supervsory Authorty, Graurhendorferstr. 108, D Bonn. The vews n ths paper should not be construed as endorsed by the GFSA.

2 nto dosyncratc and event rsk. Ths dstncton s used because events lke mergers, earnngs surprses, bankruptces and ratng mgratons are key nputs for the securty dynamcs. The separaton of market rsk nto ts general and specc components has sgncant mpact on the amount of regulatory captal requred to cover the market rsk of a tradng book. In the framework of nternal models ths captal charge s determned by means of a rsk measure, the Valueat-Rsk (VaR). Ths paper addresses ssues arsng from the applcaton of the current regulatory approach. The rch lterature on VaR comprses, for nstance, RskMetrcs (1996), Alexander (1996), Joron (000), Due & Pan (1997, 001) and Embrechts, McNeal & Straumann (1999). A sutable metrc for rsk measurement s deally derved by an adequate parsmonous modelng structure that ncorporates the essental sources of uncertanty. As prces are relatve, such a structure should dene an approprate reference unt to be used as numerare or benchmark n establshng a correspondng metrc for measurng rsk. In ths paper we suggest a benchmark approach, where we consstently use a broadly based ndex (BBI) or market wde ndex as benchmark. Ths denes a natural coordnate system n our market whch goesbeyond the regresson based beta-factor model and consderably mproves the measurement of market rsk. Furthermore, as shown n Platen (00), a BBI approxmates under general condtons the growth optmal portfolo (GOP), see Kelly (1956). Usng the GOP as reference unt the resultng benchmark model has a number of nterestng and useful propertes, see Platen (001a, 001b). We analyze n ths paper log-returns of equty prces when these are expressed n unts of the equty market ndex. Strong evdence s shown that these are Student t dstrbuted wth degrees of freedom that range typcally between 3 and 5. Ths leads to the speccaton of a Student tbenchmark model that can be shown to yeld VaR numbers consstent wth emprcal ndngs. Dscrete Tme Market Let us consder a dscrete tme equty and xed ncome market. Prces are assumed to change ther values only at the gven dscrete, equdstant tmes 0 t 0 <t 1 <:::<t n < 1 for n f0 1 :::g. The tme step sze s denoted by =t +1 ; t whch s typcally one day, f0 1 ::: n; 1g. We consder d +1 prmary assets, d f1 :::g and denote by S the value at tme t of the jth prmary securty account, whchstypcally an equty or bond wth all dvdends and coupon payments renvested. We assume throughout the paper that S > 0 (.1)

3 for all j f0 1 ::: dg and f0 1 ::: ng. We assume that S (0) s the rskless domestc savngs account at tme t, whch s a roll-over short term bond account. The return R +1 of the jth prmary securty account at tme t +1 s dened as R +1 = S +1 ; S S (.) for f0 1 ::: n;1g and j f0 1 ::: dg. The noton of a return s smple and well establshed. Equvalently, we also ntroduce the growth rato H +1 of the jth prmary securty account attmet +1 n the form H +1 = R +1 +1= S +1 S (.3) for f0 1 ::: n; 1g and j f0 1 ::: dg. Ths leads us naturally to the ntroducton of the jth log-return at tme t +1 n the form L +1 =log H +1 : (.4) Note, for typcal daly prce movements the return R +1 approxmates well the log-return L +1. For the characterzaton of a portfolo at tme t t s sucent to descrbe the vector of proportons = ( (1) 1 ::: (d) ) >, wth (;1 1) denotng the proporton of the value of the portfolo at tme t that s nvested n the jth prmary securty account, j f0 1 ::: dg. By a > we denote the transpose of a vector or a matrx. Obvously, the proportons sum to one, that s dx j=0 =1 (.5) for all f0 1 ::: ng. The value of the correspondng portfolo at tme t s denoted by S (), f0 1 ::: ng. In ths context the number j of unts of the j-th prmary securty account attmet s gven by the relaton j = S () S (.6) for j f0 1 ::: dg and f0 1 ::: ng. We obtanthegrowth rato H () ` of ths portfolo at tme t` n the form H () ` = dx j=0 `;1 H ` (.7) 3

4 whch, smlarly to (.3), can be wrtten as the portfolo return for ` f1 ::: ng. R () ` = H () ` ; 1= dx j=0 `;1 R ` (.8) 3 Regulatory Termnology and Framework Before we consder specc regulatory terms, we recall the denton of VaR. We denote by VaR h (S () ) the VaR number at tme t of a gven portfolo S () wth proportons, a gven level of sgncance and a forecast horzon of h tradng days. In practce, h s typcally chosen as one or ten tradng days, that s h f1 10g. More precsely, the VaR number VaR h (S () ) denotes the -quantle of the dstrbuton functon of the random varable S () h = S() ; ~ S () +h (3.1) where S () s known at tme t. The varable S ~ () denotes at tme +h t +h the random value at the tme t xed portfolo, that s ~S () +h = dx j=0 j S +h where the number of unts j to be held n the jth prmary securty account at tme t s gven n (.6). Typcally, the sgncance level s set to 99%. The VaR number s nterpreted as an upper bound of losses, expressed n unts of the domestc currency that mght only be surpassed wth probablty 1 ;. These losses are caused by prce changes of the underlyng securtes n the portfolo. Next we ntroduce the ocal denton of market rsk, see Basle (1996a), p.1: Market rsk s dened as the rsk of losses n on and o-balance-sheet postons arsng from movements n market prces. The rsks subject to ths requrement are: the rsks pertanng to nterest rate related nstruments and equtes n the tradng book foregn exchange rsk and commodty rsk throughout the bank. As outlned n the ntroducton, market rsk can be dvded nto general and specc market rsk. The followng denton of general market rsk s gven n Basle (1996a), p. 19 and p. 9: General market rsk covers the rsk of holdng long or short postons n nterest rate or equty rsk aganst the market as a whole. In Basle (1996a), p. ths s made more precse: The market should be dented wth a sngle factor that s representatve for the market as a whole, for example, a wdely accepted broadly based stock ndex for the country concerned. 4

5 We emphasze, t s a regulatory requrement that a broadly based ndex (BBI) serves as a reference unt. We wll naturally ncorporate ths requrement n our approach by usng a BBI as benchmark, denoted by S (). The followng denton relates specc rsk, whch ncludes event rsk, to ndces, see Basle (1996a), p. 5: Specc rsk ncludes the rsk that an ndvdual debt or equty securty moves by more orless than the general market n day-today tradng, ncludng perods when the whole market s volatle. Specc rsk covers that rsk n holdng long or short postons n an ndvdual equty or debt securty. Event rsk covers the rsk, where the prce of an ndvdual debt or equty securty moves precptously relatve to the general market due to a major event, e.g., on a take-over bd or some other shock event such events would also nclude the rsk of default. The derentaton between derent forms of rsk allows a bank to talor regulatory captal, that s captal cushons or reserves that correspond to the nherent rsks of a gven portfolo, see Basle (1996a), p.. Assume that an nternal model provdes for a portfolo S (), based on the regulatory parameters h = 10 and = 99%, the VaR number VaR 10 (S () 99%) at tme t for general market rsk. Furthermore, suppose that a prescrpton s gven to calculate separately the specc rsk wth the assocated VaR gure denoted by VaR S 10 (S() tme t for the portfolo S () C R = max 99%). In order to determne the regulatory captal C R at the followng formula has to be appled: VaR 10 S () 99% + m VaR S 10 S () 99% M 1 60 X59 `=0 VaR 10 S () ;` 99% + m 1 60 X59 l=0 VaR S 10 S () ;` 99% : Internal models that cover dosyncratc rsk but not event rsk are called surcharge models. In that case the varable m equals 1. For those models that cover specc rsk ncludng event rsk,m s set to zero. M denotes a safety multpler whch s usually set to 3. We remark, that event rsk, when t s covered by general market rsk, does not requre partcular regulatory captal. For such rsks that are related to low probablty or rare events, socalled, stress tests have to be appled, see Basle (1996a), p. 46 and Gbson (001). 4 Beta Factor Model It s common practce, see RskMetrcs (1996) and Basle (1996a), to regress the return R +1 of the jth prmary securty account at tme t +1 on the, so called, jth beta factor to separate the mpact of general and specc market rsk. To apply a beta factor model for equtes, a system of lnear 5

6 regresson equatons s then used, where R +1 = R(0) +1 + R () +1 + " +1 (4.1) for f0 1 ::: n ; 1g and j f0 1 ::: dg. Recall that R (0) +1 return of the domestc savngs account and R () +1 s the s that of the BBI. Let E denote condtonal expectaton gven the nformaton up untl tme t. Furthermore, n a beta factor model the Gaussan random varable " the jth dosyncratc nose and R () +1 are assumed to be such that E " +1 E " +1 E " +1 "(`) +1 = 0 E " +1 R() +1 = E R () " +1 = 0 E R +1 R() +1 =0 = = () () + E + R (0) +1 E R () for R () +1 for f0 1 ::: n; 1g and j ` f0 1 ::: dg wth ` 6= j. Note that the return R +1 of the jth prmary securty account n (4.1) depends lnearly on the return R () +1 of the BBI S(). The j-th dosyncratc nose " +1, s nether correlated to the market return R () +1 nor to the other dosyncratc nose terms " (k) +1 for k 6= j. As a result of these assumptons, the varance of the return of the jth prmary securty account can be decomposed nto the sum = E R +1 ; E R +1 =( ) () + (4.) " (4.3) for f0 1 ::: n; 1g and j f0 1 ::: dg, wththevarance of the return of the BBI () = E R () +1 ; E R +1 () : (4.4) In ths setup the returns and also ther varances, see (4.1) and (4.3), are lnearly related. Equaton (4.3) expresses the total market rsk of the jth prmary securty account n the form of varances. The rst and second term express the general and specc market rsk, respectvely. The above regresson based beta factor model s n accordance wth the regulatory dentons gven n Secton 3. The quantty R () +1 the so-called beta-equvalent of the return R +1 n (4.1) s. Note that relaton (4.3) s 6

7 smply a lnear decomposton of market rsk nto general and specc market rsk on the level of second moments. The beta factor model, whch s a lnear regresson model wth constant beta factor, has a purely statstcal motvaton. Addtonally, both the lnearty and Gaussan assumptons mposed on returns provde a rather crude approxmaton to observed returns. In realty, returns exhbt much fatter tals n ther dstrbutons as we wll see below. 5 Benchmark Framework In the followng, we consder an alternatve to the beta factor model. The benchmark framework allows us to separate general and specc market rsk n a canoncal way. Ths separaton s acheved n a natural settng usng a BBI S () as reference unt or benchmark. We ntroduce the jth benchmarked growth rato ^H +1 H () = H (5.1) for f0 1 ::: n; 1g and j f0 1 ::: dg, see (.3) and (.7). Equaton (5.1) allows us to express the growth rato of the jth prmary securty account as the product H +1 = H () +1 ^H +1 (5.) for f0 1 ::: n ; 1g and j f0 1 ::: dg. Ths product provdes a multplcatve decomposton of the jth growth rato. The rst factor H () ^H s related to general market rsk, whereas the second factor s naturally ted to the specc market rsk of the jth prmary securty account at tme t +1. In ths benchmark framework we denote the logarthm of the j-th benchmarked growth rato (5.1), by ^L +1 =log ^H +1 (5.3) for f0 1 ::: n; 1g and j f0 1 ::: dg. Recall that we take the tme step sze to be small. We thus obtan the condtonal expectaton E ^L +1 (5.4) whch can be assumed to be small wth some nte random varable. Note that these knd of approxmatons can be made precse n a contnuous tme settng where we let the tme step sze tend to zero. Comparng the returns of normalzed BBIs, for nstance, the S&P100, S&P500, S&P1000 and the MSCI world ndex, t s clear that these ndces behave n a very smlar manner. In Platen (00) t has been shown that 7

8 the movements of such potental BBIs approxmate those of the GOP. Thus the GOP or a proxy arses naturally when modelng general market rsk by a BBI. 6 Semparametrc Benchmark Models Let us ntroduce a general class of semparametrc benchmark models, where we assume that the jth centralzed log-return X +1 = ^L ^L +1 ; E +1 (6.1) admts the structure X +1 = ; dx k=1 p j k (k) Z +1 (6.) for j f0 1 ::: dg and f0 1 ::: n; 1g. Note that X (0) +1 s the centralzed log-return of the benchmarked domestc savngs account. Here, j k s called the jth volatlty at tme t wth respect to the kth source ofuncertanty Z (k) +1. We choose Z(1) +1 ::: Z(d) +1 as random varables wth zero mean E Z (k) +1 =0 (6.3) unt varance and such that E Z (k) +1 =1 (6.4) E Z (k) Z(`) =0 (6.5) for ` 6= k wth f0 1 ::: n; 1g and k ` f1 ::: dg. We assume for techncal reasons that an absolute moment of order slghtly greater than two exst for the vector of uncertanty Z +1 =(Z (1) +1 ::: Z(d) +1 )>. Note that n contrast to the beta factor model, we are not restrcted to the use of Gaussan random varables. Nor do we assume the ndependence of market returns and benchmarked ndvdual returns. We obtan from (6.) - (6.5) the second order normalzed condtonal moments c j ` = 1 E X +1 X (`) +1 = dx k=1 j k ` k (6.6) 8

9 for f0 1 ::: n; 1g and j ` f1 ::: dg. Relaton (6.6) allows us to ntroduce the covarance matrx as the product wth volatlty matrx =[c j ` ] d j `=1 (6.7) = D D > (6.8) D =[ j ` ] d j l=1 (6.9) f0 1 ::: n; 1g. The volatlty matrx D can be nterpreted as the Cholesky decomposton of. If the volatlty matrx D s nvertble, then, by (6.), the vector Z +1 =(Z (1) +1 ::: Z(d) +1 )> of the sources of uncertanty can be explctly expressed n the form ; 1 p D ;1 X +1 = Z +1 (6.10) usng the vector of observed growth ratos X +1 = (X (1) +1 ::: X(d) +1 )>, f0 1 ::: n; 1g. By equatons (5.1) and (6.1) wth j = 0 t can be seen that X (0) +1 = = log (0) ^L ^L(0) +1 ; E +1 H (0) +1 ; E log H (0) +1 ; log H () +1 + E log H () +1 for f0 1 ::: n; 1g. Snce the growth rato H (0) +1 of the savngs account s known at tme t the rst two terms n the above formula oset each other. Thus, we obtan from the fact that E (log(h () +1 )) () than p, the log-return of the BBI n the approxmate form log H () +1 = ;X (0) +1 + E log H () +1 s of hgher order ;X (0) +1 (6.11) for f0 1 ::: n; 1g. Ths means, the uncertanty of the log-return of the BBI s approxmately equal to the negatve of that of the benchmarked savngs account. By assumng the return R +1 of the jth prmary securty account tobe small we obtan from relaton (.3) and the expanson of the logarthm that R +1 = H +1 ; 1 log H +1 (6.1) 9

10 for j f0 1 ::: dg. Now, relaton (5.) yelds R +1 log H +1 =log H () +1 +log ^H +1 and t follows from (6.11), (5.3) and (6.1) by neglectng hgher order terms that R (0) +1 ;X +1 + X +1 : (6.13) The above descrbed semparametrc benchmark model s based on the multplcatve relatonshp (5.) between growth ratos. These naturally express general and specc market rsk. As s evdent from (6.11) and(6.), provdes a measure of the exposure of the BBI towards the k-th source of uncertanty Z (k) +1. Smlarly, by (5.3), (6.1) and (6.) the volatlty j k quantes the exposure of the j-th benchmarked prmary securty account towards the k-th source of uncertanty. One can say, the volatltes 0 k of the BBI parameterze the general market rsk, whereas the volatlty j k reects the specc market rsk of the j-th prmary securty account wth respect to the k-th source of uncertanty, f0 1 ::: n; 1g, j f0 1 ::: dg and k f1 ::: dg. Let us now provde a lnk to the beta-factor model to llustrate certan smlartes. Summarzng (6.13) and (6.) provdes the followng representaton of the stochastc component of the return of the jth prmary securty account the volatlty 0 k R +1 dx k=1 0 k p ; j k (k) Z +1 (6.14) for f0 1 ::: n; 1g and j f0 1 ::: dg. Smlarly, usng (6.11), (6.1) and (6.), we obtan for the stochastc part of the return of the BBI the approxmate expresson R () +1 = H () +1 ; 1 log H () +1 ;X (0) +1 = dx k=1 p 0 k (k) Z +1 (6.15) for f0 1 ::: n; 1g. By (6.15), (6.14), (6.3), (6.4) and (6.5) we obtan dx (6.16) E R () +1 E R +1 dx k=1 k=1 0 k 0 k ; j k (6.17) 10

11 and E R +1 R() +1 dx k=1 0 k ; j k 0 k (6.18) for f0 1 ::: n; 1g. We can dene the followng approxmate rato of covarances of returns as generalzed jth beta factor at tme t, where P E R d +1 R() +1 k=1 = 0 k ; j k P 0 k d k=1 P d =1; l=1 0 k j k P 0 l d`=1 0 ` E R () +1 for f0 1 ::: n; 1g and j f0 1 ::: dg. The above relaton provdes the equvalent of the common beta factor n a benchmark framework. Note that we get for the domestc savngs account the beta factor (0) =0,ass to be expected. Moreover, for the generalzed beta factor () of a portfolo S () wth E R () () +1 R() +1 = E R () +1 one can show by smlar arguments as above that () =1; P d k=1 0 k P d j=0 j k P d`=1 0 ` : (6.19) Note that the generalzed beta factor of the semparametrc benchmark model s tme dependent and matches the well-known return relatonshp of the Captal Asset Prcng Model, see Merton (1973). Fnally, for the GOP S () t can be shown, see Platen (001a), that dx j=0 j k 0 (6.0) for f0 1 ::: n; 1g and k f1 ::: dg. Therefore, usng (6.19) and (6.0) the generalzed beta factor () of the GOP, and thus abbi,s approxmately one. 7 Generalzed Hyperbolc Benchmark Models For the class of semparametrc benchmark models one can now specfy approprate famles of dstrbutons for the sources of uncertanty Z (1) ::: Z (d). 11

12 Let us choose the log-return dstrbuton from the well establshed class of generalzed hyperbolc dstrbutons. These dstrbutons were extensvely studed n Barndor-Nelsen (1978) and Barndor-Nelsen & Blaesld (1981). In the followng statstcal analyss we assume, for smplcty, that the centralzed log-returns of benchmarked share prces have a symmetrc generalzed hyperbolc dstrbuton. We acknowledge that a slght skewness n log-returns s typcal. However, for the small tme step sze consdered here, ths s a hgher order eect. The man feature that we explore n the followng concerns the shape of the tals of the probablty densty f X of centralzed log-returns X. The symmetrc generalzed hyperbolc densty s of the form f X (x) = 1 p K () K ; 1 r r (x ; ) (x ; )! 1 (; 1 ) (7.1) for x <, where <, 0 and = wth 6= 0 f 0 and 6= 0f 0. Here K () s the moded Bessel functon of the thrd knd wth ndex. For smplcty, we assume that the above parameters reman constant over tme. Note that the symmetrc generalzed hyperbolc dstrbuton s a four parameter famly of dstrbutons. The two shape parameters are and, dened so that they are nvarant under scale transformaton as descrbed below. The kurtoss of the log-return X s gven by the expresson X = 3 K () K + () K +1 () (7.) for ( ) [0 1) <. We remark that for =0and [; 0] we have nnte kurtoss. The parameter n (7.1) s a locaton parameter, where the log-return X has mean m X =. We dene the parameter c as the unque scale parameter such that the varance of X s v X = c and ( f =0.e., >0 =0 c = K +1() K () otherwse: It can be shown that for!1or!1the symmetrc generalzed hyperbolc dstrbuton asymptotcally approaches the Gaussan dstrbuton. Thus the log-returns of the lognormal model, see Black & Scholes (1973), appear as lmtng cases of the above class of symmetrc generalzed hyperbolc log-return dstrbutons. 1

13 We wll now descrbe four partcularly mportant symmetrc generalzed hyperbolc dstrbutons that concde wth the log-return dstrbutons of mportant asset prce models that have been suggested by derent authors: The Student t dstrbuton was orgnally suggested by Praetz (197) as a sutable dstrbuton for asset returns. The Student t dstrbuton s a specal symmetrc generalzed hyperbolc dstrbuton, where one sets = 0 and <0. The parameter = ; (7.3) s called the degrees of freedom of the Student t dstrbuton. We have nte varance for > and nte kurtoss r =3 ; for >4. The Student t dstrbuton s a three parameter dstrbuton where smaller degrees of freedom ;4 mply heaver tals. Barndor-Nelsen (1995) proposed a model, where the log-returns generate a normal-nverse Gaussan mxture dstrbuton. Ths dstrbuton appears as a specal case of the symmetrc generalzed hyperbolc dstrbuton where the shape parameter s xed at the level = ;0:5. The normalnverse Gaussan dstrbuton s also a three parameter dstrbuton. A smaller shape parameter mples larger tal heavness wth kurtoss r =3(1+ ;1 ): Eberlen & Keller (1995) and also Kuchler et al. (1995) suggested asset prce models, where log-returns are hyperbolcly dstrbuted. The hyperbolc dstrbuton results as a specal case of the symmetrc generalzed hyperbolc dstrbuton f we choose the shape parameter = 1, see Eberlen (001) for a recent survey. The hyperbolc dstrbuton s also a three parameter dstrbuton. Its kurtoss r = 3 K 1() K 3 () K () depends on the shape parameter, whch can be shown to reach ts maxmum value of r = 6 for =0. Madan & Seneta (1990), see also Geman, Madan & Yor (1998), proposed a class of models that result n log-returns whch follow a varance gamma dstrbuton. Ths dstrbuton s the specal case of the symmetrc generalzed hyperbolc dstrbuton that arses f one sets = 0 and the shape parameter strctly postve, that s > 0. A smaller mples heaver tals for ths three parameter dstrbuton wth kurtoss r = 3 (1 + ;1 ). For the lmtng case = 0 we have nnte kurtoss. 8 Testng Benchmark Models In ths secton we dentfy a dstrbuton that best ts the growth ratos of BBIs and benchmarked stock prces. We note that for several of the 13

14 aforementoned dstrbutons the kurtoss can be nnte. Therefore, asta- tstcal method that reles on hgher order moments should not be used. The maxmum lkelhood approach avods ths problem. We recall, for xed j f1 ::: dg the log-returns X, f0 1 ::: n; 1g, are ndependent and dentcally dstrbuted. We dene the lkelhood rato n the standard form = Lm L gm, where L m represents the maxmzed lkelhood functon of a partcular three parameter dstrbuton, for nstance, the Student t dstrbuton. Wth respect to ths dstrbuton the maxmum lkelhood estmate for the parameters s computed. On the other hand, L gm denotes the maxmzed lkelhood functon for the four parameter symmetrc generalzed hyperbolc dstrbuton. As n! 1 the test statstc L n = ;ln s asymptotcally chsquare dstrbuted, see Rao (1973), wth degrees of freedom one. Asymptotcally, we then have the relaton ; ; P L n <1; 1 F (1) 1; 1 =1; (8.1) as n!1, where F (1) denotes the chsquare dstrbuton wth one degree of freedom and 1; 1 ts 100(1 ; )% quantle. We can then check, say, for the standard 99% sgncance level whether or not the test statstc L n s n the 1% quantleofthechsquare dstrbuton wth one degree of freedom. Ths means, f the relaton L n < 0:01 1 0:000 (8.) s not satsed, then we reject on a 99% sgncance level the hypothess that the suggested dstrbuton s the true underlyng log-return dstrbuton. We study benchmarked US stock prce log-returns on the bass of daly benchmarked share prce data, provded by Thomson Fnancal for the eleven year perod from 1987 to 1997, usng the S&P500 as BBI. The total number of observed daly log-returns for each benchmarked stock s about 500, whch provdes relable asymptotc results for the maxmum lkelhood rato test. In Table 1 weshow the test statstcs L n for twenty leadng shares of the US market. The stock codes correspond to those commonly used. The smallest L n value dentes the best t and s dsplayed n bold type. We note that for the majorty of log-returns of benchmarked stock prces the Student t dstrbuton gves the smallest L n value and thus yelds the best t n the class of symmetrc generalzed hyperbolc dstrbutons consdered. We observe that fourteen of the twenty L n values n Table 1 appear to be smaller than the 0:01 1 =0:000 quantle. For these benchmarked stock prce log-returns the hypothess that the Student t dstrbuton s the true underlyng dstrbuton cannot be rejected at a 99% sgncance level. A smlar study has been performed for the Australan, German, Japanese and UK market usng the correspondng market ndex as BBI. Detaled results for the other four markets can be obtaned from the authors. For thrty one 14

15 Inverse Varance Stock Student t Gaussan Hyperbolc Gamma GE KO XON INTC MRK RD IBM MO PG PFE BMY T WMT JNJ LLY DD DIS HWP PEP MOB Table 1: The L n -values for log-returns of benchmarked US stocks. of the one hundred stocks consdered, the Student t dstrbuton could not be rejected at a 99% sgncance level as the true underlyng dstrbuton. In all ve markets the Student t dstrbuton clearly provdes the best t for most log-returns of the benchmarked stocks. To llustrate the results for all ve markets we plot n Fgure 1 the estmated ( )-parameter values for the one hundred examned benchmarked stocks. These estmates characterze the specc shape of the tals for the estmated symmetrc generalzed hyperbolc denstes. The postve partof the -axs n Fgure 1 parameterzes the varance gamma dstrbuton. We note that only one of the one hundred stocks generated an ( )-estmate on the postve -axs. Consequently, thevarance gamma dstrbuton does not t our data well. The hyperbolc dstrbuton s represented by pars of shape parameters ( ) wth =1. There are about three to four ponts n Fgure 1 that are located n the neghborhood of the horzontal lne =1. Thus the t of the hyperbolc dstrbuton s also qute lmted. The ponts ( ) on the horzontal lne = ; 1 correspond to the normal-nverse Gaussan dstrbuton. We note that several stocks generate ponts n the area close to ths lne. Therefore, the normal-nverse Gaussan dstrbuton provdes a 15

16 reasonable t to the data. We remark that for small the normal-nverse Gaussan dstrbuton concdes asymptotcally wth the Student t dstrbuton and most parameter estmates are less than one. It s obvous that most of the one hundred ( )-estmates are concentrated close to the negatve -axs. Ths s the area that s characterstc for dstrbutons that are smlar to the Student t dstrbuton. It appears that the ( )-estmates support a Student t dstrbuton wth a parameter value for of approxmately ;. Ths corresponds to a Student t dstrbuton of four degrees of freedom, see (7.3). The average estmated degrees of freedom for the Student t dstrbuton obtaned from all benchmarked stocks was ^ =4: lambda alphabar Fgure 1: ( )-plot for log-returns of benchmarked stocks. In a study smlar to that descrbed above t has been shown n Hurst & Platen (1997) that the log-returns of the BBIs of the equty markets of most developed economes, ncludng the ve also consdered here, support the Student t dstrbuton, agan wth degrees of freedom close to four. As an alternatve to the above mentoned studes, a seres of papers, see, for nstance, Dacarogna et al. (1995), has shown that the tals of asset log-returns follow approxmately a power law, where the estmates of the so-called tal ndex are typcally n the range of three to ve. Ths s consstent wth our ndngs for benchmarked stock log-returns. Furthermore, we remark that the Student t benchmark model can be nterpreted as a dscrete tme analog of the, socalled, mnmal market model, see Platen (001a, 001b), whch models the dynamcs of a GOP that exhbts mnmum varance n ts drft. These lead to Student t dstrbuted log-returns of the GOP wth exactly = 4 degrees of freedom. 16

17 9 VaR Analyss As outlned n the ntroducton and n Secton 3, the modelng of event rsk n nternal models s of ncreasng mportance n VaR analyss. The above class of semparametrc benchmark models s able to encompass event rskby choosng an adequate leptokurtc dstrbuton for the sources of uncertanty. As prevously explaned, the Student t dstrbuton provdes an excellenttto log-returns of benchmarked equty data. Incorporatng event rsk completes VaR modelng for measurng market rsk by takng nto account all regulatory subcategores of market rsk, that s general, dosyncratc and event rsk. As an alternatve to the above approach, Gbson (001) used a mxng dstrbuton to specfy a ve parameter model that puts sucent mass nto the tals of log-return dstrbutons by ntroducng derent regmes for the means of certan mxed dstrbutons. Due& Pan (001) apply jump-dusons for modelng specc and event rsk n the context of VaR whch reles on varous parameters. Though the class of jump-dusons s ntutvely appealng, the parameters are dcult to estmate, see Honore (1998). Huschens & Km (1999) studed a VaR model that uses Student t dstrbuted returns but not n a benchmark settng. Note that we specfy n a parsmonous way the sources of uncertanty n the followng verson of the Student tbenchmark model. We explot the fact that symmetrc generalzed hyperbolc dstrbutons admt a representaton of a mxture of normal dstrbutons. Ths means, f one chooses the varance of a condtonally Gaussan dstrbuton as the nverse of a Gamma dstrbuted random varable, then the resultng dstrbuton s a Student t dstrbuton. To generate the Student t dstrbuted sources of uncertanty Z (0) +1, :::, Z(d) +1 approprately, weset where +1 denotes the market actvty wth +1 = 1 ; 1 X Z (k) +1 = p +1 Y (k) +1 (9.1) `=1 (`) +1! ;1 (9.) for f0 1 ::: n; 1g and k f1 ::: dg wth f3 4 :::g. Addtonally to the ndependent standard Gaussan random varables Y (k) +1 that appear n (9.1) we employ further ndependent standard Gaussan random varables (`) +1. Hence, the random varables +1 are chsquare dstrbuted wth degrees of freedom. Then the random varables Z (k) +1 are Student t dstrbuted wth unt varance and degrees of freedom. The market actvty +1 can be nterpreted as a daly ncrement of the random busness tme of the market. Note, the market actvty converges to 1 as the degrees of freedom tend to nnty, whch yelds the lognormal benchmark model. 17

18 In addton to the typcal parameters of the lognormal benchmark model we have used here only the extra parameter, whch s sucent tocharac- terze the leptokurtoss of the Student t dstrbuton. As shown prevously, the typcal parameter choce for s about 4. Smaller degrees of freedom generate log-returns wth more extreme movements. An mportant feature of the resultng multvarate Student t dstrbuton s ts copula. It realstcally captures the dependence of extreme asset prce movements as shown n Embrechts, McNeal & Straumann (1999) and related work. In our model the Student t-copula, characterzng the jont occurrence of extreme moves n log-returns of several benchmarked stocks, s automatcally obtaned. To calbrate the above benchmark Student t model one needs to estmate the volatltes j k,typcally obtaned from a standard lognormal calbraton. The above model then represents a smple extenson of the lognormal model, where one needs only to add an estmate for the degrees of freedom. It s reasonable to set =4wthout losng much accuracy, as we wll see from Table. The equatons (9.1) and (9.) gve aprescrpton that can also be used for smulaton purposes. They nvolve the Cholesky decomposton D of the covarance matrx, see (6.9). The Student t benchmark model provdes qute accurate results f the VaR calculaton s based on extensve Monte- Carlo smulatons. However, reasonably accurate Monte-Carlo smulatons are extremely tme consumng. To crcumvent ths problem, n the followng we propose a hghly ecent method for VaR calculatons. In practce, equty portfolos, are typcally domnated by large lnear portfolos. Our model s n that case analytcally tractable. Note that the jont ::: X(d) +1 )> s a multvar- dstrbuton of the random vector X +1 =(X (1) +1 ate Student t dstrbuton wth degrees of freedom, where Snce Y +1 s Gaussan and 1 +1 X +1 = p +1 D Y +1 : s ndependent chsquare dstrbuted, the resultng multvarate Student t dstrbuton for X +1 belongs to the class of ellptcal dstrbutons. For lnear portfolos the calculaton of VaR numbers s for ths class of dstrbutons analytcally tractable. More precsely, a theorem n Fang, Kotz & Ng (1990) yelds the representaton a > X +1 = ka > D k +1 (9.3) for any gven weght vector a, where kk s the Eucldean norm, a < d, D > D = and +1 denotes a Student t dstrbuted scalar random varable wth degrees of freedom. The representaton (9.3) sgncantly smples the VaR calculaton even for lnear portfolos wth an extremely large number of consttuents. Furthermore, we menton that f a portfolo wth many consttuents s not lnear but represents a dversed portfolo n the sense 18

19 descrbed by Platen (00), then t approxmates the GOP and thus also our benchmark, the BBI. These asymptotcs yeld accurate VaR numbers and save computatonal tme when compared to standard Monte-Carlo smulaton. Snce the multvarate Student t dstrbuton s an ellptcal dstrbuton, t s shown by Embrechts, McNeal & Straumann (1999), that VaR s n ths case a coherent rsk measure, see Artzner et al. (1997). Thus the crtcal subaddtvty ofvar for the Student tbenchmark model s guaranteed. Ths fact s hghly mportant for the consstent use of VaR as an nternal rsk measure. Snce t allows the nternal captal allocaton to partcular busness lnes, the Student t benchmark model oers sgncant support for nternal rsk management. In order to calculate VaR for short term horzons we apply the so-called square root tme rule. Ths approxmaton s n lne wth the regulatory recommendatons of the Basle Commttee. From (9.3) we obtan then the followng formula for the VaR number of a gven portfolo S () at tme t. VaR h (S () ) V pa > a p h ~ t (): (9.4) Here V denotes the market value of the portfolo at tme t, p a > a characterzes the volatlty of the portfolo, the tme step sze for a tradng day, h the number of tradng days and ~ t () the-quantle of the standardzed Student t dstrbuton wth degrees of freedom. Obvously, the product (9.4) generalzes the well-known short hand formula, used n RskMetrcs, to calculate VaR when one uses the event factor that s ' = ~ t () q (9.5) VaR h (S () ) V pa > aq a p h ' (9.6) where q s the -quantle of the standard Gaussan dstrbuton. Consequently, the event factor' adjusts the standard VaR formula to a level that captures approxmately event rsk when one uses the Student t benchmark model as nternal model. Accordng to the quantles of the Gaussan and the Student t dstrbuton one obtans by (9.4) the event factors shown n Table ' Table : Event factor ' n dependence on degrees of freedom. Even for rather small degrees of freedom, say, the addtonal regulatory captal wll not surpass 16%. To conrm the approprate choce of the model 19

20 and ts calbraton one needs to perform stress tests. Along these lnes, Gbson (001) performed an extensve study usng representatve portfolos for US nsttutons, whch show that the event factor should be close to 1:1. In other words, by means of extensve hstorcal smulaton Gbson (001) dented an event factor of ' =1:1. Ths s the event factor that exactly matches n Table the degrees of freedom =4,whch strongly supports our emprcal ndng of approxmately four degrees of freedom for log-returns. References Alexander, C. (1996). Handbook of Rsk Management and Analyss. Wley, Chchester. Artzner, P., F. Delbaen, J. M. Eber, & D. Heath (1997). Thnkng coherently. Rsk 10, 68{71. Barndor-Nelsen, O. (1978). Hyperbolc dstrbutons and dstrbutons on hyperbolae. Scandnavan Journal of Statstcs 5, 151{157. Barndor-Nelsen, O. (1995). NormalnnInverse Gaussan processes and the modellng of stock returns. Techncal report, Unversty of Aarhus Barndor-Nelsen, O. & P. Blaesld (1981). Hyperbolc dstrbutons and ramcatons: Contrbutons to theory and applcaton. In C. Talle, G. P. Patl, and B. A. Baldessar (Eds.), Statstcal Dstrbutons n Scentc Work, Volume 4, pp. 19{44. Dordrecht: Redel. Basle (1996a). Amendment to the Captal Accord to Incorporate Market Rsks. Basle Commttee on Bankng and Supervson, Basle, Swtzerland. Basle (1996b). Modcatons to the Market Rsk Amendment. Basle Commttee on Bankng and Supervson, Basle, Swtzerland. Black, F. & M. Scholes (1973). The prcng of optons and corporate labltes. J. Poltcal Economy 81, 637{659. Dacarogna, M., U. A. Muller, O. V. Pctet, & C. G. De Vres (1995). The dstrbuton of extremal foregn exchange rate returns n extremely large data sets. A preprnt of the Olsen Group, Zurch. Due, D. &J.Pan (1997). An overvew of Value at Rsk. J. of Dervatves 4(3), 7{49. Due, D. & J. Pan (001). Analytc Value at Rsk wth jumps and credt rsk. Fnance and Stochastcs 5, 155{180. Eberlen, E. (001). Applcatons of generalzed Levy motons n nance. In O. E. Barndor-Nelsen, T. Mkosch, and S. Resnck (Eds.), Levy Processes: Theory and Applcaton. Eberlen, E. & U. Keller (1995). Hyperbolc dstrbutons n nance. Bernoull 1, 81{99. 0

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo

More information

Vasicek s Model of Distribution of Losses in a Large, Homogeneous Portfolio

Vasicek s Model of Distribution of Losses in a Large, Homogeneous Portfolio Vascek s Model of Dstrbuton of Losses n a Large, Homogeneous Portfolo Stephen M Schaefer London Busness School Credt Rsk Electve Summer 2012 Vascek s Model Important method for calculatng dstrbuton of

More information

Analysis of Premium Liabilities for Australian Lines of Business

Analysis of Premium Liabilities for Australian Lines of Business Summary of Analyss of Premum Labltes for Australan Lnes of Busness Emly Tao Honours Research Paper, The Unversty of Melbourne Emly Tao Acknowledgements I am grateful to the Australan Prudental Regulaton

More information

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ). REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or

More information

An Alternative Way to Measure Private Equity Performance

An Alternative Way to Measure Private Equity Performance An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate

More information

Course outline. Financial Time Series Analysis. Overview. Data analysis. Predictive signal. Trading strategy

Course outline. Financial Time Series Analysis. Overview. Data analysis. Predictive signal. Trading strategy Fnancal Tme Seres Analyss Patrck McSharry patrck@mcsharry.net www.mcsharry.net Trnty Term 2014 Mathematcal Insttute Unversty of Oxford Course outlne 1. Data analyss, probablty, correlatons, vsualsaton

More information

Portfolio Loss Distribution

Portfolio Loss Distribution Portfolo Loss Dstrbuton Rsky assets n loan ortfolo hghly llqud assets hold-to-maturty n the bank s balance sheet Outstandngs The orton of the bank asset that has already been extended to borrowers. Commtment

More information

The Application of Fractional Brownian Motion in Option Pricing

The Application of Fractional Brownian Motion in Option Pricing Vol. 0, No. (05), pp. 73-8 http://dx.do.org/0.457/jmue.05.0..6 The Applcaton of Fractonal Brownan Moton n Opton Prcng Qng-xn Zhou School of Basc Scence,arbn Unversty of Commerce,arbn zhouqngxn98@6.com

More information

Recurrence. 1 Definitions and main statements

Recurrence. 1 Definitions and main statements Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.

More information

Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.

Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001. Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.

More information

Simon Acomb NAG Financial Mathematics Day

Simon Acomb NAG Financial Mathematics Day 1 Why People Who Prce Dervatves Are Interested In Correlaton mon Acomb NAG Fnancal Mathematcs Day Correlaton Rsk What Is Correlaton No lnear relatonshp between ponts Co-movement between the ponts Postve

More information

DI Fund Sufficiency Evaluation Methodological Recommendations and DIA Russia Practice

DI Fund Sufficiency Evaluation Methodological Recommendations and DIA Russia Practice DI Fund Suffcency Evaluaton Methodologcal Recommendatons and DIA Russa Practce Andre G. Melnkov Deputy General Drector DIA Russa THE DEPOSIT INSURANCE CONFERENCE IN THE MENA REGION AMMAN-JORDAN, 18 20

More information

Can Auto Liability Insurance Purchases Signal Risk Attitude?

Can Auto Liability Insurance Purchases Signal Risk Attitude? Internatonal Journal of Busness and Economcs, 2011, Vol. 10, No. 2, 159-164 Can Auto Lablty Insurance Purchases Sgnal Rsk Atttude? Chu-Shu L Department of Internatonal Busness, Asa Unversty, Tawan Sheng-Chang

More information

The Cox-Ross-Rubinstein Option Pricing Model

The Cox-Ross-Rubinstein Option Pricing Model Fnance 400 A. Penat - G. Pennacc Te Cox-Ross-Rubnsten Opton Prcng Model Te prevous notes sowed tat te absence o arbtrage restrcts te prce o an opton n terms o ts underlyng asset. However, te no-arbtrage

More information

The Analysis of Outliers in Statistical Data

The Analysis of Outliers in Statistical Data THALES Project No. xxxx The Analyss of Outlers n Statstcal Data Research Team Chrysses Caron, Assocate Professor (P.I.) Vaslk Karot, Doctoral canddate Polychrons Economou, Chrstna Perrakou, Postgraduate

More information

Copulas. Modeling dependencies in Financial Risk Management. BMI Master Thesis

Copulas. Modeling dependencies in Financial Risk Management. BMI Master Thesis Copulas Modelng dependences n Fnancal Rsk Management BMI Master Thess Modelng dependences n fnancal rsk management Modelng dependences n fnancal rsk management 3 Preface Ths paper has been wrtten as part

More information

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of

More information

Fixed income risk attribution

Fixed income risk attribution 5 Fxed ncome rsk attrbuton Chthra Krshnamurth RskMetrcs Group chthra.krshnamurth@rskmetrcs.com We compare the rsk of the actve portfolo wth that of the benchmark and segment the dfference between the two

More information

Pricing index options in a multivariate Black & Scholes model

Pricing index options in a multivariate Black & Scholes model Prcng ndex optons n a multvarate Black & Scholes model Danël Lnders AFI_1383 Prcng ndex optons n a multvarate Black & Scholes model Danël Lnders Verson: October 2, 2013 1 Introducton In ths paper, we consder

More information

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange

More information

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..

More information

On the pricing of illiquid options with Black-Scholes formula

On the pricing of illiquid options with Black-Scholes formula 7 th InternatonalScentfcConferenceManagngandModellngofFnancalRsks Ostrava VŠB-TU Ostrava, Faculty of Economcs, Department of Fnance 8 th 9 th September2014 On the prcng of llqud optons wth Black-Scholes

More information

CHAPTER 14 MORE ABOUT REGRESSION

CHAPTER 14 MORE ABOUT REGRESSION CHAPTER 14 MORE ABOUT REGRESSION We learned n Chapter 5 that often a straght lne descrbes the pattern of a relatonshp between two quanttatve varables. For nstance, n Example 5.1 we explored the relatonshp

More information

Calculation of Sampling Weights

Calculation of Sampling Weights Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a two-stage stratfed cluster desgn. 1 The frst stage conssted of a sample

More information

Pricing Multi-Asset Cross Currency Options

Pricing Multi-Asset Cross Currency Options CIRJE-F-844 Prcng Mult-Asset Cross Currency Optons Kenchro Shraya Graduate School of Economcs, Unversty of Tokyo Akhko Takahash Unversty of Tokyo March 212; Revsed n September, October and November 212

More information

1 De nitions and Censoring

1 De nitions and Censoring De ntons and Censorng. Survval Analyss We begn by consderng smple analyses but we wll lead up to and take a look at regresson on explanatory factors., as n lnear regresson part A. The mportant d erence

More information

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12

PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12 14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed

More information

Measuring portfolio loss using approximation methods

Measuring portfolio loss using approximation methods Scence Journal of Appled Mathematcs and Statstcs 014; (): 4-5 Publshed onlne Aprl 0, 014 (http://www.scencepublshnggroup.com/j/sjams) do: 10.11648/j.sjams.01400.11 Measurng portfolo loss usng approxmaton

More information

The covariance is the two variable analog to the variance. The formula for the covariance between two variables is

The covariance is the two variable analog to the variance. The formula for the covariance between two variables is Regresson Lectures So far we have talked only about statstcs that descrbe one varable. What we are gong to be dscussng for much of the remander of the course s relatonshps between two or more varables.

More information

Statistical Methods to Develop Rating Models

Statistical Methods to Develop Rating Models Statstcal Methods to Develop Ratng Models [Evelyn Hayden and Danel Porath, Österrechsche Natonalbank and Unversty of Appled Scences at Manz] Source: The Basel II Rsk Parameters Estmaton, Valdaton, and

More information

ENTERPRISE RISK MANAGEMENT IN INSURANCE GROUPS: MEASURING RISK CONCENTRATION AND DEFAULT RISK

ENTERPRISE RISK MANAGEMENT IN INSURANCE GROUPS: MEASURING RISK CONCENTRATION AND DEFAULT RISK ETERPRISE RISK MAAGEMET I ISURACE GROUPS: MEASURIG RISK COCETRATIO AD DEFAULT RISK ADIE GATZERT HATO SCHMEISER STEFA SCHUCKMA WORKIG PAPERS O RISK MAAGEMET AD ISURACE O. 35 EDITED BY HATO SCHMEISER CHAIR

More information

Construction Rules for Morningstar Canada Target Dividend Index SM

Construction Rules for Morningstar Canada Target Dividend Index SM Constructon Rules for Mornngstar Canada Target Dvdend Index SM Mornngstar Methodology Paper October 2014 Verson 1.2 2014 Mornngstar, Inc. All rghts reserved. The nformaton n ths document s the property

More information

NON-CONSTANT SUM RED-AND-BLACK GAMES WITH BET-DEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia

NON-CONSTANT SUM RED-AND-BLACK GAMES WITH BET-DEPENDENT WIN PROBABILITY FUNCTION LAURA PONTIGGIA, University of the Sciences in Philadelphia To appear n Journal o Appled Probablty June 2007 O-COSTAT SUM RED-AD-BLACK GAMES WITH BET-DEPEDET WI PROBABILITY FUCTIO LAURA POTIGGIA, Unversty o the Scences n Phladelpha Abstract In ths paper we nvestgate

More information

ECONOMICS OF PLANT ENERGY SAVINGS PROJECTS IN A CHANGING MARKET Douglas C White Emerson Process Management

ECONOMICS OF PLANT ENERGY SAVINGS PROJECTS IN A CHANGING MARKET Douglas C White Emerson Process Management ECONOMICS OF PLANT ENERGY SAVINGS PROJECTS IN A CHANGING MARKET Douglas C Whte Emerson Process Management Abstract Energy prces have exhbted sgnfcant volatlty n recent years. For example, natural gas prces

More information

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by 6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng

More information

Addendum to: Importing Skill-Biased Technology

Addendum to: Importing Skill-Biased Technology Addendum to: Importng Skll-Based Technology Arel Bursten UCLA and NBER Javer Cravno UCLA August 202 Jonathan Vogel Columba and NBER Abstract Ths Addendum derves the results dscussed n secton 3.3 of our

More information

Exchange rate volatility and its impact on risk management with internal models in commercial banks

Exchange rate volatility and its impact on risk management with internal models in commercial banks Banks and Bank Systems, Volume, Issue 4, 007 Devjak Sreko (Slovena), Andraž Grum (Slovena) Exchange rate volatlty and ts mpact on rsk management wth nternal models n commercal banks Abstract Fnancal markets

More information

HOUSEHOLDS DEBT BURDEN: AN ANALYSIS BASED ON MICROECONOMIC DATA*

HOUSEHOLDS DEBT BURDEN: AN ANALYSIS BASED ON MICROECONOMIC DATA* HOUSEHOLDS DEBT BURDEN: AN ANALYSIS BASED ON MICROECONOMIC DATA* Luísa Farnha** 1. INTRODUCTION The rapd growth n Portuguese households ndebtedness n the past few years ncreased the concerns that debt

More information

BERNSTEIN POLYNOMIALS

BERNSTEIN POLYNOMIALS On-Lne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful

More information

Transition Matrix Models of Consumer Credit Ratings

Transition Matrix Models of Consumer Credit Ratings Transton Matrx Models of Consumer Credt Ratngs Abstract Although the corporate credt rsk lterature has many studes modellng the change n the credt rsk of corporate bonds over tme, there s far less analyss

More information

A Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy S-curve Regression

A Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy S-curve Regression Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy S-curve Regresson Cheng-Wu Chen, Morrs H. L. Wang and Tng-Ya Hseh Department of Cvl Engneerng, Natonal Central Unversty,

More information

Efficient Project Portfolio as a tool for Enterprise Risk Management

Efficient Project Portfolio as a tool for Enterprise Risk Management Effcent Proect Portfolo as a tool for Enterprse Rsk Management Valentn O. Nkonov Ural State Techncal Unversty Growth Traectory Consultng Company January 5, 27 Effcent Proect Portfolo as a tool for Enterprse

More information

Macro Factors and Volatility of Treasury Bond Returns

Macro Factors and Volatility of Treasury Bond Returns Macro Factors and Volatlty of Treasury Bond Returns Jngzh Huang Department of Fnance Smeal Colleage of Busness Pennsylvana State Unversty Unversty Park, PA 16802, U.S.A. Le Lu School of Fnance Shangha

More information

L10: Linear discriminants analysis

L10: Linear discriminants analysis L0: Lnear dscrmnants analyss Lnear dscrmnant analyss, two classes Lnear dscrmnant analyss, C classes LDA vs. PCA Lmtatons of LDA Varants of LDA Other dmensonalty reducton methods CSCE 666 Pattern Analyss

More information

Underwriting Risk. Glenn Meyers. Insurance Services Office, Inc.

Underwriting Risk. Glenn Meyers. Insurance Services Office, Inc. Underwrtng Rsk By Glenn Meyers Insurance Servces Offce, Inc. Abstract In a compettve nsurance market, nsurers have lmted nfluence on the premum charged for an nsurance contract. hey must decde whether

More information

Financial Mathemetics

Financial Mathemetics Fnancal Mathemetcs 15 Mathematcs Grade 12 Teacher Gude Fnancal Maths Seres Overvew In ths seres we am to show how Mathematcs can be used to support personal fnancal decsons. In ths seres we jon Tebogo,

More information

x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60

x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60 BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true

More information

What is Candidate Sampling

What is Candidate Sampling What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble

More information

Dscrete-Tme Approxmatons of the Holmstrom-Mlgrom Brownan-Moton Model of Intertemporal Incentve Provson 1 Martn Hellwg Unversty of Mannhem Klaus M. Schmdt Unversty of Munch and CEPR Ths verson: May 5, 1998

More information

The OC Curve of Attribute Acceptance Plans

The OC Curve of Attribute Acceptance Plans The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4

More information

A Model of Private Equity Fund Compensation

A Model of Private Equity Fund Compensation A Model of Prvate Equty Fund Compensaton Wonho Wlson Cho Andrew Metrck Ayako Yasuda KAIST Yale School of Management Unversty of Calforna at Davs June 26, 2011 Abstract: Ths paper analyzes the economcs

More information

ErrorPropagation.nb 1. Error Propagation

ErrorPropagation.nb 1. Error Propagation ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then

More information

Staff Paper. Farm Savings Accounts: Examining Income Variability, Eligibility, and Benefits. Brent Gloy, Eddy LaDue, and Charles Cuykendall

Staff Paper. Farm Savings Accounts: Examining Income Variability, Eligibility, and Benefits. Brent Gloy, Eddy LaDue, and Charles Cuykendall SP 2005-02 August 2005 Staff Paper Department of Appled Economcs and Management Cornell Unversty, Ithaca, New York 14853-7801 USA Farm Savngs Accounts: Examnng Income Varablty, Elgblty, and Benefts Brent

More information

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy 4.02 Quz Solutons Fall 2004 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.

More information

Exhaustive Regression. An Exploration of Regression-Based Data Mining Techniques Using Super Computation

Exhaustive Regression. An Exploration of Regression-Based Data Mining Techniques Using Super Computation Exhaustve Regresson An Exploraton of Regresson-Based Data Mnng Technques Usng Super Computaton Antony Daves, Ph.D. Assocate Professor of Economcs Duquesne Unversty Pttsburgh, PA 58 Research Fellow The

More information

Forecasting the Direction and Strength of Stock Market Movement

Forecasting the Direction and Strength of Stock Market Movement Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract - Stock market s one of the most complcated systems

More information

IN THE UNITED STATES THIS REPORT IS AVAILABLE ONLY TO PERSONS WHO HAVE RECEIVED THE PROPER OPTION RISK DISCLOSURE DOCUMENTS.

IN THE UNITED STATES THIS REPORT IS AVAILABLE ONLY TO PERSONS WHO HAVE RECEIVED THE PROPER OPTION RISK DISCLOSURE DOCUMENTS. http://mm.pmorgan.com European Equty Dervatves Strategy 4 May 005 N THE UNTED STATES THS REPORT S AVALABLE ONLY TO PERSONS WHO HAVE RECEVED THE PROPER OPTON RS DSCLOSURE DOCUMENTS. Correlaton Vehcles Technques

More information

1. Measuring association using correlation and regression

1. Measuring association using correlation and regression How to measure assocaton I: Correlaton. 1. Measurng assocaton usng correlaton and regresson We often would lke to know how one varable, such as a mother's weght, s related to another varable, such as a

More information

How Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence

How Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence 1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh

More information

SPECIALIZED DAY TRADING - A NEW VIEW ON AN OLD GAME

SPECIALIZED DAY TRADING - A NEW VIEW ON AN OLD GAME August 7 - August 12, 2006 n Baden-Baden, Germany SPECIALIZED DAY TRADING - A NEW VIEW ON AN OLD GAME Vladmr Šmovć 1, and Vladmr Šmovć 2, PhD 1 Faculty of Electrcal Engneerng and Computng, Unska 3, 10000

More information

A Probabilistic Theory of Coherence

A Probabilistic Theory of Coherence A Probablstc Theory of Coherence BRANDEN FITELSON. The Coherence Measure C Let E be a set of n propostons E,..., E n. We seek a probablstc measure C(E) of the degree of coherence of E. Intutvely, we want

More information

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.

Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt. Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces

More information

An Analysis of Pricing Methods for Baskets Options

An Analysis of Pricing Methods for Baskets Options An Analyss of Prcng Methods for Baskets Optons Martn Krekel, Johan de Kock, Ralf Korn, Tn-Kwa Man Fraunhofer ITWM, Department of Fnancal Mathematcs, 67653 Kaserslautern, Germany, emal: krekel@twm.fhg.de

More information

The Development of Web Log Mining Based on Improve-K-Means Clustering Analysis

The Development of Web Log Mining Based on Improve-K-Means Clustering Analysis The Development of Web Log Mnng Based on Improve-K-Means Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.

More information

Lecture 3: Force of Interest, Real Interest Rate, Annuity

Lecture 3: Force of Interest, Real Interest Rate, Annuity Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annuty-mmedate, and ts present value Study annuty-due, and

More information

THE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES

THE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered

More information

Brigid Mullany, Ph.D University of North Carolina, Charlotte

Brigid Mullany, Ph.D University of North Carolina, Charlotte Evaluaton And Comparson Of The Dfferent Standards Used To Defne The Postonal Accuracy And Repeatablty Of Numercally Controlled Machnng Center Axes Brgd Mullany, Ph.D Unversty of North Carolna, Charlotte

More information

CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol

CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK Sample Stablty Protocol Background The Cholesterol Reference Method Laboratory Network (CRMLN) developed certfcaton protocols for total cholesterol, HDL

More information

Cautiousness and Measuring An Investor s Tendency to Buy Options

Cautiousness and Measuring An Investor s Tendency to Buy Options Cautousness and Measurng An Investor s Tendency to Buy Optons James Huang October 18, 2005 Abstract As s well known, Arrow-Pratt measure of rsk averson explans a ratonal nvestor s behavor n stock markets

More information

Reporting Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (including SME Corporate), Sovereign and Bank Instruction Guide

Reporting Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (including SME Corporate), Sovereign and Bank Instruction Guide Reportng Forms ARF 113.0A, ARF 113.0B, ARF 113.0C and ARF 113.0D FIRB Corporate (ncludng SME Corporate), Soveregn and Bank Instructon Gude Ths nstructon gude s desgned to assst n the completon of the FIRB

More information

Hedging Interest-Rate Risk with Duration

Hedging Interest-Rate Risk with Duration FIXED-INCOME SECURITIES Chapter 5 Hedgng Interest-Rate Rsk wth Duraton Outlne Prcng and Hedgng Prcng certan cash-flows Interest rate rsk Hedgng prncples Duraton-Based Hedgng Technques Defnton of duraton

More information

ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING

ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING ANALYZING THE RELATIONSHIPS BETWEEN QUALITY, TIME, AND COST IN PROJECT MANAGEMENT DECISION MAKING Matthew J. Lberatore, Department of Management and Operatons, Vllanova Unversty, Vllanova, PA 19085, 610-519-4390,

More information

Project Networks With Mixed-Time Constraints

Project Networks With Mixed-Time Constraints Project Networs Wth Mxed-Tme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa

More information

DEFINING %COMPLETE IN MICROSOFT PROJECT

DEFINING %COMPLETE IN MICROSOFT PROJECT CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMI-SP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,

More information

Stress test for measuring insurance risks in non-life insurance

Stress test for measuring insurance risks in non-life insurance PROMEMORIA Datum June 01 Fnansnspektonen Författare Bengt von Bahr, Younes Elonq and Erk Elvers Stress test for measurng nsurance rsks n non-lfe nsurance Summary Ths memo descrbes stress testng of nsurance

More information

Time Value of Money. Types of Interest. Compounding and Discounting Single Sums. Page 1. Ch. 6 - The Time Value of Money. The Time Value of Money

Time Value of Money. Types of Interest. Compounding and Discounting Single Sums. Page 1. Ch. 6 - The Time Value of Money. The Time Value of Money Ch. 6 - The Tme Value of Money Tme Value of Money The Interest Rate Smple Interest Compound Interest Amortzng a Loan FIN21- Ahmed Y, Dasht TIME VALUE OF MONEY OR DISCOUNTED CASH FLOW ANALYSIS Very Important

More information

On the Optimal Control of a Cascade of Hydro-Electric Power Stations

On the Optimal Control of a Cascade of Hydro-Electric Power Stations On the Optmal Control of a Cascade of Hydro-Electrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal;

More information

Risk-based Fatigue Estimate of Deep Water Risers -- Course Project for EM388F: Fracture Mechanics, Spring 2008

Risk-based Fatigue Estimate of Deep Water Risers -- Course Project for EM388F: Fracture Mechanics, Spring 2008 Rsk-based Fatgue Estmate of Deep Water Rsers -- Course Project for EM388F: Fracture Mechancs, Sprng 2008 Chen Sh Department of Cvl, Archtectural, and Envronmental Engneerng The Unversty of Texas at Austn

More information

Intra-year Cash Flow Patterns: A Simple Solution for an Unnecessary Appraisal Error

Intra-year Cash Flow Patterns: A Simple Solution for an Unnecessary Appraisal Error Intra-year Cash Flow Patterns: A Smple Soluton for an Unnecessary Apprasal Error By C. Donald Wggns (Professor of Accountng and Fnance, the Unversty of North Florda), B. Perry Woodsde (Assocate Professor

More information

The Short-term and Long-term Market

The Short-term and Long-term Market A Presentaton on Market Effcences to Northfeld Informaton Servces Annual Conference he Short-term and Long-term Market Effcences en Post Offce Square Boston, MA 0209 www.acadan-asset.com Charles H. Wang,

More information

Implied (risk neutral) probabilities, betting odds and prediction markets

Implied (risk neutral) probabilities, betting odds and prediction markets Impled (rsk neutral) probabltes, bettng odds and predcton markets Fabrzo Caccafesta (Unversty of Rome "Tor Vergata") ABSTRACT - We show that the well known euvalence between the "fundamental theorem of

More information

Forecasting the Demand of Emergency Supplies: Based on the CBR Theory and BP Neural Network

Forecasting the Demand of Emergency Supplies: Based on the CBR Theory and BP Neural Network 700 Proceedngs of the 8th Internatonal Conference on Innovaton & Management Forecastng the Demand of Emergency Supples: Based on the CBR Theory and BP Neural Network Fu Deqang, Lu Yun, L Changbng School

More information

Calculating the high frequency transmission line parameters of power cables

Calculating the high frequency transmission line parameters of power cables < ' Calculatng the hgh frequency transmsson lne parameters of power cables Authors: Dr. John Dcknson, Laboratory Servces Manager, N 0 RW E B Communcatons Mr. Peter J. Ncholson, Project Assgnment Manager,

More information

2.4 Bivariate distributions

2.4 Bivariate distributions page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together

More information

MAPP. MERIS level 3 cloud and water vapour products. Issue: 1. Revision: 0. Date: 9.12.1998. Function Name Organisation Signature Date

MAPP. MERIS level 3 cloud and water vapour products. Issue: 1. Revision: 0. Date: 9.12.1998. Function Name Organisation Signature Date Ttel: Project: Doc. No.: MERIS level 3 cloud and water vapour products MAPP MAPP-ATBD-ClWVL3 Issue: 1 Revson: 0 Date: 9.12.1998 Functon Name Organsaton Sgnature Date Author: Bennartz FUB Preusker FUB Schüller

More information

The Cross Section of Foreign Currency Risk Premia and Consumption Growth Risk

The Cross Section of Foreign Currency Risk Premia and Consumption Growth Risk The Cross Secton of Foregn Currency Rsk Prema and Consumpton Growth Rsk By HANNO LUSTIG AND ADRIEN VERDELHAN* Aggregate consumpton growth rsk explans why low nterest rate currences do not apprecate as

More information

A Simplified Framework for Return Accountability

A Simplified Framework for Return Accountability Reprnted wth permsson from Fnancal Analysts Journal, May/June 1991. Copyrght 1991. Assocaton for Investment Management and Research, Charlottesvlle, VA. All rghts reserved. by Gary P. Brnson, Bran D. Snger

More information

Evaluating credit risk models: A critique and a new proposal

Evaluating credit risk models: A critique and a new proposal Evaluatng credt rsk models: A crtque and a new proposal Hergen Frerchs* Gunter Löffler Unversty of Frankfurt (Man) February 14, 2001 Abstract Evaluatng the qualty of credt portfolo rsk models s an mportant

More information

Stochastic epidemic models revisited: Analysis of some continuous performance measures

Stochastic epidemic models revisited: Analysis of some continuous performance measures Stochastc epdemc models revsted: Analyss of some contnuous performance measures J.R. Artalejo Faculty of Mathematcs, Complutense Unversty of Madrd, 28040 Madrd, Span A. Economou Department of Mathematcs,

More information

ESTIMATING THE MARKET VALUE OF FRANKING CREDITS: EMPIRICAL EVIDENCE FROM AUSTRALIA

ESTIMATING THE MARKET VALUE OF FRANKING CREDITS: EMPIRICAL EVIDENCE FROM AUSTRALIA ESTIMATING THE MARKET VALUE OF FRANKING CREDITS: EMPIRICAL EVIDENCE FROM AUSTRALIA Duc Vo Beauden Gellard Stefan Mero Economc Regulaton Authorty 469 Wellngton Street, Perth, WA 6000, Australa Phone: (08)

More information

Time Series Analysis in Studies of AGN Variability. Bradley M. Peterson The Ohio State University

Time Series Analysis in Studies of AGN Variability. Bradley M. Peterson The Ohio State University Tme Seres Analyss n Studes of AGN Varablty Bradley M. Peterson The Oho State Unversty 1 Lnear Correlaton Degree to whch two parameters are lnearly correlated can be expressed n terms of the lnear correlaton

More information

Latent Class Regression. Statistics for Psychosocial Research II: Structural Models December 4 and 6, 2006

Latent Class Regression. Statistics for Psychosocial Research II: Structural Models December 4 and 6, 2006 Latent Class Regresson Statstcs for Psychosocal Research II: Structural Models December 4 and 6, 2006 Latent Class Regresson (LCR) What s t and when do we use t? Recall the standard latent class model

More information

Interest Rate Fundamentals

Interest Rate Fundamentals Lecture Part II Interest Rate Fundamentals Topcs n Quanttatve Fnance: Inflaton Dervatves Instructor: Iraj Kan Fundamentals of Interest Rates In part II of ths lecture we wll consder fundamental concepts

More information

v a 1 b 1 i, a 2 b 2 i,..., a n b n i.

v a 1 b 1 i, a 2 b 2 i,..., a n b n i. SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are

More information

Trade Adjustment and Productivity in Large Crises. Online Appendix May 2013. Appendix A: Derivation of Equations for Productivity

Trade Adjustment and Productivity in Large Crises. Online Appendix May 2013. Appendix A: Derivation of Equations for Productivity Trade Adjustment Productvty n Large Crses Gta Gopnath Department of Economcs Harvard Unversty NBER Brent Neman Booth School of Busness Unversty of Chcago NBER Onlne Appendx May 2013 Appendx A: Dervaton

More information

The impact of bank capital requirements on bank risk: an econometric puzzle and a proposed solution

The impact of bank capital requirements on bank risk: an econometric puzzle and a proposed solution Banks and Bank Systems, Volume 4, Issue 1, 009 Robert L. Porter (USA) The mpact of bank captal requrements on bank rsk: an econometrc puzzle and a proposed soluton Abstract The relatonshp between bank

More information

7.5. Present Value of an Annuity. Investigate

7.5. Present Value of an Annuity. Investigate 7.5 Present Value of an Annuty Owen and Anna are approachng retrement and are puttng ther fnances n order. They have worked hard and nvested ther earnngs so that they now have a large amount of money on

More information

CS 2750 Machine Learning. Lecture 3. Density estimation. CS 2750 Machine Learning. Announcements

CS 2750 Machine Learning. Lecture 3. Density estimation. CS 2750 Machine Learning. Announcements Lecture 3 Densty estmaton Mlos Hauskrecht mlos@cs.ptt.edu 5329 Sennott Square Next lecture: Matlab tutoral Announcements Rules for attendng the class: Regstered for credt Regstered for audt (only f there

More information

Binomial Link Functions. Lori Murray, Phil Munz

Binomial Link Functions. Lori Murray, Phil Munz Bnomal Lnk Functons Lor Murray, Phl Munz Bnomal Lnk Functons Logt Lnk functon: ( p) p ln 1 p Probt Lnk functon: ( p) 1 ( p) Complentary Log Log functon: ( p) ln( ln(1 p)) Motvatng Example A researcher

More information

Chapter 2 The Basics of Pricing with GLMs

Chapter 2 The Basics of Pricing with GLMs Chapter 2 The Bascs of Prcng wth GLMs As descrbed n the prevous secton, the goal of a tarff analyss s to determne how one or more key ratos Y vary wth a number of ratng factors Ths s remnscent of analyzng

More information